PX155 - D9 - driven damped simple harmonic motion

add an additional oscillatory force $F = F_{0} \cos ω t$ to the damped mass on a spring, where $ω =$ applied frequency
PX155 - A2 - newton's second law: $m \ddot{x} = - k x - b \dot{x} + F_{0} \cos ω t$
- dividing by $m$ and switching to complex notation:
  $\ddot{z} + γ \dot{z} + ω_{0}^{2} z = \frac{F_{0}}{m} e^{i ω t} . . . [1]$
- note: $ω_{0} = \sqrt{\frac{k}{m}}$ , $ω$ is the frequency of the applied oscillating force
we care about long term solutions, after damped SHM solutions have died away
try: $z = a e^{i ω t}$
$\dot{z} = i ω z$
$\ddot{z} = - ω^{2} z$
in [1] : $- ω^{2} z + i ω γ z + ω_{0}^{2} z = \frac{F_{0}}{m} e^{i ω t} . . . [1]$
$(- ω^{2} + i ω γ + ω_{0}^{2}) a e^{i ω t} = \frac{F_{0}}{m} e^{i ω t}$
$a = \frac{F / m}{(ω_{0}^{2} - ω^{2}) + i ω γ}$ this is a complex number
recall $a = | a | e^{i ϕ}$ in modulus argument form
$z = | a | e^{i (ω t + ϕ)}$
- with $x = R e (z) = | a | \cos (ω t + ϕ)$
equate two forms of $a$ :

\frac{F / m}{(ω_{0}^{2} - ω^{2}) + i ω γ} = | a | e^{i ϕ}

B2: $| a | = \frac{F / m}{\sqrt{(ω_{0}^{2} - ω^{2})^{2} - (ω γ)^{2}}}$ , $\tan ϕ = \frac{- γ ω}{ω_{0}^{2} - ω^{2}}$
comparison: consider a static load ( $W = 0$ )
$| a |_{0} = \frac{F_{0} / m}{ω_{0}^{2}} = \frac{F_{0}}{k}$
- $| a |_{0}$ is an extension of spring under static load:

\frac{| a |}{| a |_{0}} = \frac{ω_{0}^{2}}{\sqrt{(ω_{0}^{2} - ω^{2}) + (γ ω)^{2}}}

- for small $ω_{0} - ω$ and small $γ$ , we can have $\frac{| a |}{| a |_{0}} > 1$ , so, large amplitude and static load